Theorem mulnqprlemru 7658

Description: Lemma for mulnqpr 7661. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemru	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem mulnqprlemru

Dummy variables f g h r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7631	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7631	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-imp 7553	. . . . . . 7 |- .P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g .Q h ) ) } >. )
4		mulclnq 7460	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
5	3, 4	genpelvu 7597	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
6	1, 2, 5	syl2an 289	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
7	6	biimpa 296	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) )
8		vex 2766	. . . . . . . . . . . . 13 |- s e. _V
9		breq2 4038	. . . . . . . . . . . . 13 |- ( u = s -> ( A <Q u <-> A <Q s ) )
10		ltnqex 7633	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7634	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op2nd 6214	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
13	8, 9, 12	elab2 2912	. . . . . . . . . . . 12 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q s )
14	13	biimpi 120	. . . . . . . . . . 11 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q s )
15	14	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> A <Q s )
16	15	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> A <Q s )
17		vex 2766	. . . . . . . . . . . . 13 |- t e. _V
18		breq2 4038	. . . . . . . . . . . . 13 |- ( u = t -> ( B <Q u <-> B <Q t ) )
19		ltnqex 7633	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7634	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op2nd 6214	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) = { u | B <Q u }
22	17, 18, 21	elab2 2912	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> B <Q t )
23	22	biimpi 120	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) -> B <Q t )
24	23	ad2antll 491	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> B <Q t )
25	24	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> B <Q t )
26		ltrelnq 7449	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4716	. . . . . . . . . . 11 |- ( A <Q s -> ( A e. Q. /\ s e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A e. Q. /\ s e. Q. ) )
29	26	brel 4716	. . . . . . . . . . 11 |- ( B <Q t -> ( B e. Q. /\ t e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( B e. Q. /\ t e. Q. ) )
31		lt2mulnq 7489	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ s e. Q. ) /\ ( B e. Q. /\ t e. Q. ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
32	28, 30, 31	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
33	16, 25, 32	mp2and 433	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q ( s .Q t ) )
34		breq2 4038	. . . . . . . . 9 |- ( r = ( s .Q t ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
36	33, 35	mpbird 167	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q r )
37		vex 2766	. . . . . . . 8 |- r e. _V
38		breq2 4038	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
39		ltnqex 7633	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
40		gtnqex 7634	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
41	39, 40	op2nd 6214	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
42	37, 38, 41	elab2 2912	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
43	36, 42	sylibr 134	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
44	43	ex 115	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2622	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
46	7, 45	mpd 13	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
47	46	ex 115	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
48	47	ssrdv 3190	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182   E.wrex 2476   C_ wss 3157   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   2ndc2nd 6206   Q.cnq 7364   .Q cmq 7367   <Q cltq 7369   P.cnp 7375   .P. cmp 7378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-inp 7550  df-imp 7553

This theorem is referenced by:  mulnqprlemfl  7659  mulnqpr  7661